Showing papers on "Nonlinear system published in 1971"

Polynomial Theory of Complex Systems

Abstract: A complex multidimensional decision hypersurface can be approximated by a set of polynomials in the input signals (properties) which contain information about the hypersurface of interest. The hypersurface is usually described by a number of experimental (vector) points and simple functions of their coordinates. The approach taken in this paper to approximating the decision hypersurface, and hence the input-output relationship of a complex system, is to fit a high-degree multinomial to the input properties using a multilayered perceptronlike network structure. Thresholds are employed at each layer in the network to identify those polynomials which best fit into the desired hypersurface. Only the best combinations of the input properties are allowed to pass to succeeding layers, where more complex combinations are formed. Each element in each layer in the network implements a nonlinear function of two inputs. The coefficients of each element are determined by a regression technique which enables each element to approximate the true outputs with minimum mean-square error. The experimental data base is divided into a training and testing set. The training set is used to obtain the element coefficients, and the testing set is used to determine the utility of a given element in the network and to control overfitting of the experimental data. This latter feature is termed "decision regularization.

Light waves in thin films and integrated optics.

Abstract: Integrated optics is a far-reaching attempt to apply thin-film technology to optical circuits and devices, and, by using methods of integrated circuitry, to achieve a better and more economical optical system. The specific topics discussed here are physics of light waves in thin films, materials and losses involved, methods of couplings light beam into and out of a thin film, and nonlinear interactions in waveguide structures. The purpose of this paper is to review in some detail the important development of this new and fascinating field, and to caution the reader that the technology involved is difficult because of the smallness and perfection demanded by thin-film optical devices.

Generation of semi-groups of nonlinear transformations on general banach spaces,

Abstract: : The authors continue a discussion of a problem posed by Hille (1951) in a paper titled, 'On the Generation of Semigroups and the Theory of Conjugate Functions.'

Simultaneous Numerical Solution of Differential-Algebraic Equations

Abstract: A unified method for handling the mixed differential and algebraic equations of the type that commonly occur in the transient analysis of large networks or in continuous system simulation is discussed. The first part of the paper is a brief review of existing techniques of handling initial value problems for stiff ordinary differential equations written in the standard form y' f(y, t) . In the second part one of these techniques is applied to the problem F(y, y', t)=0 . This may be either a differential or an algebraic equation as \partial F/ \partial y' is nonzero or zero. It will represent a mixed system when vectors F and y represent components of a system. The method lends itself to the use of sparse matrix techniques when the problem is sparse.

Simulation of Multivariate and Multidimensional Random Processes

Abstract: Efficient and practical methods of simulating multivariate and multidimensional processes with specified cross‐spectral density are presented. When the cross‐spectral density matrix of an n‐variate process is specified, its component processes can be simulated as the sum of cosine functions with random frequencies and random phase angles. Typical examples of this type are the simulation, for the purpose of shaker test, of a multivariate process representing six components of the acceleration (due to, for example, a booster engine cutoff) measured at the base of a spacecraft and the simulation of horizontal and vertical components of earthquake acceleration. A homogeneous multidimensional process can also be simulated in terms of the sum of cosine functions with random frequencies and random phase angles. Examples of multidimensional processes considered here include the horizontal component f0(t,x) of the wind velocity perpendicular to the axis (x axis) of a slender structure, the vertical gust velocity field f0(x,y) frozen in space, and the boundary‐layer pressure field f0(x,y,t). Also, a convenient use of the present method of simulation in a class of nonlinear structural vibration analysis is described with a numerical example.

Inertial navigation systems analysis

Abstract: This volume offers the avionic systems engineer a fundamental exposition of the mechanization and error analysis of inertial navigation systems. While the material is applicable to spacecraft and undersea navigation, emphasis is placed upon terrestrial applications on or slightly above the earth's surface. As a result, practical considerations are geared toward those aircraft navigation systems of particular current interest. Extensive use is made of perturbation techniques to develop linearized system equations, whose solutions closely approximate those obtained by nonlinear differential equations. A unified error analysis technique is developed that is applicable to virtually all system configurations. The technique provides a greatly simplified method for comparing the performance of competing system configurations.

The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs

Abstract: Troublesome distortions often occur in communication systems. For a wide class of systems such distortions can be computed with the help of Volterra series. Results, both old and new, which will aid the reader in applying Volterra-series-type analyses to systems driven by sine waves or Gaussian noise are presented. The n-fold Fourier transform G n of the nth Volterra kernel plays an important role in the analysis. Methods of computing G n from the system equations are described and several special systems are considered. When the G n are known, items of interest regarding the output can be obtained by substituting the G n in general formulas derived from the Volterra series representation. These items include expressions for the output harmonics, when the input is the sum of two or three sine waves, and the power spectrum and various moments, when the input is Gaussian. Special attention is paid to the case in which the Volterra series consists of only the linear and quadratic terms.

The Analysis of Feedback Systems

Abstract: "Parts of this monograph appeared in the author's doctoral dissertation entitled 'Nonlinear harmonic analysis' 1968"

Atmospheric Predictability and Two-Dimensional Turbulence

Abstract: Recent observations indicate that the planetary-scale motions of the atmosphere obey some of the laws of two-dimensional turbulence. The eddy-damped Markovian approximation to two-dimensional turbulence is applied to these motions to predict for an observed energy spectrum the nonlinear transfer rates, characteristic error spectra, and the rate of error growth. In this way estimates are derived of the predictability of the atmosphere and of the errors inherent in numerical models. The use of stochastic models for turbulence approximations is described in an Appendix.

A non-linear instability theory for a wave system in plane Poiseuille flow

Abstract: The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are et and e½(x+a1rt), where t is the time, x the distance in the direction of flow, e the growth rate of linearized theory and (−a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.

Monotonicity Methods in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations

Abstract: Publisher Summary This chapter discusses the monotonicity methods in Hilbert spaces and presents some applications to nonlinear partial differential equations. It describes classical properties of maximal monotone operators in Hilbert spaces. It focuses on a particular class of monotone operators, namely those that are gradients of convex functions. The chapter also highlights their specific properties that do not hold for general monotone operators. Evolution equations associated with gradients of convex functions: smoothing effect on the initial data, behavior at infinity, and so on are discussed in the chapter along with some applications to nonlinear partial differential equations.

Optimal adaptive estimation: Structure and parameter adaption

Abstract: Optimal structure and parameter adaptive estimators have been obtained for continuous as well as discrete data Gaussian process models with linear dynamics. Specifically, the essentially nonlinear adaptive estimators are shown to be decomposable (partition theorem) into two parts, a linear nonadaptive part consisting of a bank of Kalman-Bucy filters and a nonlinear part that incorporates the adaptive nature of the estimator. The conditional-error-covariance matrix of the estimator is also obtained in a form suitable for on-line performance evaluation. The adaptive estimators are applied to the problem of state estimation with non-Gaussian initial state, to estimation under measurement uncertainty (joint detection-estimation) as well as to system identification. Examples are given of the application of the adaptive estimators to structure and parameter adaptation indicating their applicability to engineering problems.

Viscoelastic Characterization of a Nonlinear Fiber-Reinforced Plastic

Abstract: The nonlinear, viscoelastic behavior of a unidirectional, glass fiber-epoxy composite material is characterized by using isothermal, uniaxial creep and recovery tests together with a constitutive equation based on thermodynamic theory. The nonlinear constitutive equation for uniaxial loading is described first, and then fourth-order tensor transformations relating principal linear viscoelastic creep compli ances, uniaxial creep compliance, and fiber angle are summarized. Following a discussion of experimental aspects, creep and recovery data obtained from several different specimens (each having a differ ent fiber orientation relative to the loading axis) are reduced using a graphical shifting procedure and tensor transformations to evaluate all material properties, including the principal creep compliances. As a check on the constitutive theory, the data are shown to be in ternally consistent. Some simplicity in the analytical representation of the data is found; viz. the nonlinear, uniaxial creep compli...

RMS Envelope Equations with Space Charge

Abstract: Envelope equations for a continuous beam with uniform charge density and elliptical cross-section were first derived by Kapchinsky and Vladimirsky (K-V). In fact, the K-V equations are not restricted to uniformly charged beams, but are equally valid for any charge distribution with elliptical symmetry, provided the beam boundary and emittance are defined by rms (root-meansquare) values. This results because (i) the second moments of any particle distribution depend only on the linear part of the force (determined by least squares method), while (ii) this linear part of the force in turn depends only on the second moments of the distribution. This is also true in practice for three-dimensional bunched beams with ellipsoidal symmetry, and allows the formulation of envelope equations that include the effect of space charge on bunch length and energy spread. The utility of this rms approach was first demonstrated by Lapostolle for stationary distributions. Subsequently, Gluckstern proved that the rms version of the K-V equations remain valid for all continuous beams with axial symmetry. In this report these results are extended to continuous beams with elliptical symmetry as well as to bunched beams with ellipsoidal form, and also to one-dimensional motion.

Nonlinear Theory of Random Vibrations

Abstract: Publisher Summary Random vibration analysis of mechanical systems has become an important subject in recent years, principally because of advances in high speed flight. To design structures and equipment that will survive the randomly fluctuating loads caused by the flow of turbulent air or the efflux of jet or rocket engines, it has become necessary to develop a theory capable of analyzing the effect of such fluctuating loads on structures and equipment. Many of the techniques developed for the analysis of random excitation of nonlinear control systems are applicable to the analysis of nonlinear random vibrations, and conversely many of the techniques developed in the theory of nonlinear random vibrations are equally applicable to problems in communication theory and electronics. The chapter presents modeling in nonlinear random vibrations by Markov processes. The chief reason for adopting the idealized model of a system of differential equations excited by white noise is that the computations are much simpler in this case. One of the difficulties involved in modeling nonlinear random vibrations by Markov processes is that—one is restricted to quasi-linear systems. In the subsequent development of the theory, no distinction has been made between the physical nonlinearity and the mathematical model of that nonlinearity. Further, the chapter also discusses the basic theory of stochastic processes and its applications and solution techniques.

Parallel Propagation of Nonlinear Low‐Frequency Waves in High‐β Plasma

Abstract: A pair of coupled, nonlinear, partial differential equations which describe the evolution of low‐frequency, large‐scale‐length perturbations propagating parallel, or nearly parallel, to the equilibrium magnetic field in high‐β plasma have been obtained. The equations account for irreversible resonant particle effects. In the regime of small but finite propagation angles, the pair of equations collapses into a single Korteweg‐de Vries equation (neglecting irreversible terms) which agrees with known results.

The motion of floating bodies

Abstract: We shall restrict ourselves here to floating bodies without any means of propelling themselves. The body may, of course, be a ship lying dead in the water, but there is no real limitation to practical shapes of any particular sort except that we shall suppose the body to be hydrostatically stable. This will restrict the extent of this survey in an important way: we are able to slough off all effects associated with an average velocity of the body. Since mathematical solution of problems almost inevitably proceeds by way of linearization of the boundary conditions, this means that we may avoid introducing a linearization parameter whose smallness expresses the fact that the body doesn't disturb the water much during its forward motion, for example, slenderness or thinness. If we do introduce such a geometrical assumption, it will be an additional approximation, not one forced upon us by the physical situation. Fortunately, Newman's (1970) article treats, among other things, the recent advances in the theory of motion of slender ships under way. More can be found in a paper by Ogilvie (1964) . We shall assume from the beginning that motions are small and take this into account in formulating equations and boundary conditions. Further­ more, we shaH assume the fluid inviscid, and without surface tension. It is not difficult to write down equations and boundary conditions for a less restricted problem. However, since most results are for the case of small motions and since the perturbation expansions associated with the deriva­ tion of the linearized problem from the more exact one do not present any special points of interest, it seems more efficient to start with the simpler problem. Even so, some account will be given of recent attempts to consider nonlinear problems.

Non-Linear Growth and Limiting Amplitude of Acoustic Oscillations in Combustion Chambers

Abstract: Due to non-linear loss or gain of energy, unstable oscillations in combustion chambers cannot grow indefinitely. Very often the limiting amplitudes are sufficiently low that the wave motions appear to be sinusoidal without discontinuities. The analysis presented here is based on the idea that the gasdynamics throughout most of the volume can be handled in a linear fashion. Non-linear behavior is associated with localized energy losses, such as wall losses and particle attenuation, or with the interaction between the oscillations and the combustion processes which sustain the motions. The formal procedure describes the non-linear growth and decay of an acoustic mode whose spatial structure does not change with time. Integration of the conservation equations over the volume of the chamber produces a single non-linear ordinary differential equation for the time-dependent amplitude of the mode. The equation can be solved easily by standard techniques, producing very simple results for the non-linear growth rate, decay rate, and limiting amplitude. Most of the treatment is developed for unstable motions in solid propellant rocket chambers. Other combustion chambers can be represented as special cases of the general description.

Amplitude Limitation of a Collisional Drift Wave Instability

Abstract: A nonlinear analysis of collisional drift waves is presented in which a systematic expansion is made in powers of the wave amplitude. The two‐fluid equations are used, including the effects of resistivity, viscosity, and thermal transport. The result, for the wave amplitude as a function of magnetic field in the linearly unstable region close to marginal stability, agrees reasonably well with experiment.

Isotropic solutions of the Einstein-Boltzmann equations

Abstract: It is shown that in all solutions of the Einstein-Boltzmann equations in which the particle distribution function is isotropic about some 4-velocity field, the distortion of that velocity field vanishes; further, either its expansion or its rotation vanishes. We discuss briefly further kinetic solutions in which the energy-momentum tensor has a perfect fluid form.

Overland Flow on an Infiltrating Surface

Abstract: The partial differential equation for vertical, one-phase, unsaturated moisture flow in soils is employed as a mathematical model for infiltration rate. Solution of this equation for the rainfall-ponding upper boundary condition is proposed as a sensitive means to describe infiltration rate as a dependent upper boundary condition. A nonlinear Crank-Nicholson implicit finite difference scheme is used to develop a solution to this equation that predicts infiltration under realistic upper boundary and soil matrix conditions. The kinematic wave approximation to the equations of unsteady overland flow on cascaded planes is solved by a second order explicit difference scheme. The difference equations of infiltration and overland flow are then combined into a model for a simple watershed that employs computational logic so that boundary conditions match at the soil surface. The mathematical model is tested by comparison with data from a 40-foot laboratory soil flume fitted with a rainfall simulator and with data from the USDA Agricultural Research Service experimental watershed at Hastings, Nebraska. Good agreement is obtained between measured and predicted hydrographs, although there are some differences in recession lengths. The results indicate that a theoretically based model can be used to describe simple watershed response when appropriate physical parameters can be obtained.

An innovations approach to least-squares estimation--Part III: Nonlinear estimation in white Gaussian noise

Abstract: In Parts I and II of this paper, we presented the innovations approach to linear least-squares estimation in additive white noise. In the present paper, we show how to extend this technique to the nonlinear estimation (filtering and smoothing) of non-Gaussian signals in additive white Gaussian noise. The use of the innovations allows us to obtain formulas and simple derivations that are remarkably similar to those used for the linear case thereby distinguishing clearly the essential points at which the nonlinear problem differs from the linear one.